The superfluid density, a spectral weight contributing to superconducting condensates,
determines the fundamental nature of superconductivity, such as phase stiffness [1]},
and thus closely pertains to phase fluctuations.
A quantum spin liquid (QSL) is a nontrivial magnetic state free from order and possibly bearing
fractionalization, entanglement and/or chirality [2]}, [3]}, [4]}.
The superconductivity that emerges by doping such an anomalous magnetic state
[5]}, [6]}, [7]}, [8]}, [9]} is of keen interest with
regard to the unconventional pairing nature out of a BCS framework.
In the absence of adequate model materials that address this question,
the layered organic conductor with a triangular lattice,
\(\kappa \) -(BEDT-TTF)\(_{4}\) Hg\(_{2.89}\) Br\(_{8}\)  (\(\kappa \) -HgBr) [10]},
has recently arisen as a promising doped QSL candidate.
The present work experimentally investigates the nature of superconductivity that emerges
in this system in terms of the superfluid density,
which is a key parameter to elucidate the possible unconventionality of pairing.
